Just as in our recent 3-photon calculation [1]}, we have decomposed the amplitudes using the helicity projection method proposed in ref. [2]}. The construction of the gluon/photon polarization vector follows closely ref. [1]}. The fermion projector in our previous work was designed with the permutation symmetry of the final-state photons in mind. The partonic processes considered here are less symmetric and thus we consider a simpler projector. In the case of the \(q\bar{q} \rightarrow g \gamma \gamma \)  partonic process, the fermionic line is projected by considering the following trace in spin space:
\(\mathcal {M} = \bar{v}(p_2,h_2) \Gamma u(p_1,h_1)= \biggl \lbrace  \left(u \otimes \bar{v}\right)\Gamma \biggr \rbrace \,,\) 
